Vortex is nothing but rotationally of flow on their own axis vorticity is talked in individuality but eddies are talked in plural means many of vortices are eddies
Example of vorticity is whirlpool of water in washing machine it has no gap between them but in case of eddies there is a gap between them ex is separation of flow on the airfoil in eddies back motion is created as you had seen on the airfoil
swril is like eddies in this there is a creation of backmotion and building of gap
Eddies is created due to generation of swril
In vortex motion the flow rotates on its own axis but does not translates
Turbulence is basically the chaos eddies and swirling produces turbulence
Vortex is present both in laminar and turbulence region but it's nature changes or shape changes
Swirling are very popular for fuel mixing in engines there are swirl ratio and tumblr ratio which decide the mixing of fluid how good they mix
Vortex is fluid rotating about an axis and it can be of vortex tubes or vortex bundles or infinite vortex or vortex sheet.
Swirl is a flow with angular momentum. The flow axially moves and also rotates and spread about the axis along which it moves.
Eddies are blobs of fluids looking like ball and rotates in arbitrary directions as dictated by the flow gradients and they can break up to form smaller eddies or merge up to form larger eddies.